Subgroups — C₆₇×#59666 #30738917

Order 17152.

1596 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2802	C₂	1	4	yes	maximal
2	C₆₇×#2582	C₂	1	2	yes	maximal
3	C₆₇×#2581	C₂	1	2	yes	maximal
4	C₆₇×#2571	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2565	C₂	1	2	yes	maximal
6	C₆₇×#2552	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#2550	C₂	1	2	yes	maximal
8	C₆₇×#2520	C₂	1	4	yes	maximal
9	C₆₇×#2508	C₂	1	2	yes	maximal
10	C₆₇×#2507	C₂	1	2	yes	maximal
11	C₆₇×#2370	C₂	1	2	yes	maximal
12	C₆₇×#2365	C₂	1	1	yes	maximal, characteristic
13	C₆₇×#2365	C₂	1	1	yes	maximal, characteristic
14	C₆₇×#2351	C₂	1	2	yes	maximal
15	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
16	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
17	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#125×C₆₇	C₂²	1	2	yes	—
21	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	#400×C₆₇	C₂²	1	4	yes	—
27	#400×C₆₇	C₂²	1	4	yes	—
28	#398×C₆₇	C₂²	1	4	yes	—
29	#398×C₆₇	C₂²	1	4	yes	—
30	#398×C₆₇	C₂²	1	4	yes	—
31	#398×C₆₇	C₂²	1	4	yes	—
32	#398×C₆₇	C₂²	1	4	yes	—
33	#395×C₆₇	C₂²	1	4	yes	—
34	#391×C₆₇	C₂²	1	2	yes	—
35	#391×C₆₇	C₂²	1	4	yes	—
36	#391×C₆₇	C₂²	1	2	yes	—
37	#391×C₆₇	C₂²	1	2	yes	—
38	#391×C₆₇	C₂²	1	2	yes	—
39	#389×C₆₇	C₂²	1	2	yes	—
40	#389×C₆₇	C₂²	1	2	yes	—
41	#389×C₆₇	C₂²	1	4	yes	—
42	#389×C₆₇	C₂²	1	2	yes	—
43	#389×C₆₇	C₂²	1	2	yes	—
44	#389×C₆₇	C₂²	1	1	yes	characteristic
45	#389×C₆₇	C₂²	1	2	yes	—
46	#389×C₆₇	C₂²	1	2	yes	—
47	#389×C₆₇	C₂²	1	2	yes	—
48	#389×C₆₇	C₂²	1	4	yes	—
49	#389×C₆₇	C₂²	1	4	yes	—
50	#389×C₆₇	C₂²	1	2	yes	—
51	#389×C₆₇	C₂²	1	1	yes	characteristic
52	#389×C₆₇	C₂²	1	1	yes	characteristic
53	#389×C₆₇	C₂²	1	1	yes	characteristic
54	#389×C₆₇	C₂²	1	2	yes	—
55	#387×C₆₇	C₂²	1	4	yes	—
56	#387×C₆₇	C₂²	1	4	yes	—
57	#387×C₆₇	C₂²	1	2	yes	—
58	#387×C₆₇	C₂²	1	4	yes	—
59	#387×C₆₇	C₂²	1	4	yes	—
60	#387×C₆₇	C₂²	1	4	yes	—
61	#387×C₆₇	C₂²	1	2	yes	—
62	#387×C₆₇	C₂²	1	4	yes	—
63	#384×C₆₇	C₂²	1	1	yes	characteristic
64	#384×C₆₇	C₂²	1	1	yes	characteristic
65	#384×C₆₇	C₂²	1	2	yes	—
66	#382×C₆₇	C₂²	1	4	yes	—
67	#382×C₆₇	C₂²	1	2	yes	—
68	#382×C₆₇	C₂²	1	1	yes	characteristic
69	#382×C₆₇	C₂²	1	2	yes	—
70	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
74	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
75	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
76	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
77	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
78	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
79	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
80	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
81	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
82	C₄³×C₆₇	C₂²	1	1	yes	characteristic
83	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
84	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
85	C₄⋊Q₈×C₆₇	—	2	4	no	—
86	C₄⋊Q₈×C₆₇	Q₈	1	8	yes	complemented
87	#125×C₆₇	—	2	8	no	—
88	#125×C₆₇	—	2	4	no	—
89	C₄×Q₈×C₆₇	—	2	8	no	—
90	C₄×Q₈×C₆₇	—	4	4	no	—
91	C₄×Q₈×C₆₇	—	4	4	no	—
92	C₄×Q₈×C₆₇	—	4	4	no	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
119	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
124	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
136	#95×C₆₇	C₂³	1	2	yes	—
137	#95×C₆₇	C₂³	1	2	yes	—
138	#95×C₆₇	C₂³	1	2	yes	—
139	#95×C₆₇	C₂³	1	2	yes	—
140	#95×C₆₇	C₂³	1	2	yes	—
141	#95×C₆₇	C₂³	1	2	yes	—
142	#95×C₆₇	C₂³	1	2	yes	—
143	#95×C₆₇	C₂³	1	4	yes	—
144	#95×C₆₇	C₂³	1	4	yes	—
145	#95×C₆₇	C₂³	1	2	yes	—
146	#95×C₆₇	C₂³	1	2	yes	—
147	#95×C₆₇	C₂³	1	4	yes	—
148	#95×C₆₇	C₂³	1	4	yes	—
149	#95×C₆₇	C₂³	1	2	yes	—
150	#95×C₆₇	C₂³	1	4	yes	—
151	#95×C₆₇	C₂³	1	4	yes	—
152	#95×C₆₇	C₂³	1	4	yes	—
153	#95×C₆₇	C₂³	1	4	yes	—
154	#95×C₆₇	C₂³	1	4	yes	—
155	#95×C₆₇	C₂³	1	4	yes	—
156	C₂×Q₈×C₆₇	—	2	4	no	—
157	C₂×Q₈×C₆₇	—	2	4	no	—
158	C₂×Q₈×C₆₇	—	2	4	no	—
159	C₂×Q₈×C₆₇	—	2	4	no	—
160	C₂×Q₈×C₆₇	—	4	2	no	—
161	C₂×Q₈×C₆₇	C₂×Q₈	1	8	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
179	C₄⋊C₄×C₆₇	—	2	8	no	—
180	C₄⋊C₄×C₆₇	—	2	4	no	—
181	C₄⋊C₄×C₆₇	—	2	8	no	—
182	C₄⋊C₄×C₆₇	—	2	8	no	—
183	C₄⋊C₄×C₆₇	—	2	4	no	—
184	C₄⋊C₄×C₆₇	—	2	4	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	2	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	—	4	2	no	—
196	C₄⋊C₄×C₆₇	—	4	2	no	—
197	C₄⋊C₄×C₆₇	—	4	2	no	—
198	C₄⋊C₄×C₆₇	—	4	4	no	—
199	C₄⋊C₄×C₆₇	—	4	4	no	—
200	C₄⋊C₄×C₆₇	—	4	2	no	—
201	C₄⋊C₄×C₆₇	—	4	2	no	—
202	C₄⋊C₄×C₆₇	—	4	4	no	—
203	C₄⋊C₄×C₆₇	—	4	4	no	—
204	C₄⋊C₄×C₆₇	—	4	2	no	—
205	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
206	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
207	C₄⋊C₄×C₆₇	#41	1	4	yes	—
208	C₄²×C₆₇	—	2	8	no	—
209	C₄²×C₆₇	—	2	4	no	—
210	C₄²×C₆₇	—	2	2	no	—
211	C₄²×C₆₇	—	2	4	no	—
212	C₄²×C₆₇	—	2	2	no	—
213	C₄²×C₆₇	—	2	4	no	—
214	C₄²×C₆₇	—	2	2	no	—
215	C₄²×C₆₇	—	2	2	no	—
216	C₄²×C₆₇	—	2	4	no	—
217	C₄²×C₆₇	—	4	1	no	—
218	C₄²×C₆₇	—	4	4	no	—
219	C₄²×C₆₇	—	4	4	no	—
220	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
221	C₄²×C₆₇	#41	1	4	yes	—
222	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
223	Q₈×C₆₇	—	2	16	no	—
224	Q₈×C₆₇	—	4	8	no	—
225	C₂×C₄×C₆₇	—	2	1	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	1	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	1	no	—
230	C₂×C₄×C₆₇	—	2	4	no	—
231	C₂×C₄×C₆₇	—	2	1	no	—
232	C₂×C₄×C₆₇	—	2	1	no	—
233	C₂×C₄×C₆₇	—	2	1	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	1	no	—
236	C₂×C₄×C₆₇	—	2	1	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	1	no	—
252	C₂×C₄×C₆₇	—	2	2	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	2	no	—
257	C₂×C₄×C₆₇	—	2	2	no	—
258	C₂×C₄×C₆₇	—	2	1	no	—
259	C₂×C₄×C₆₇	—	2	4	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
265	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
266	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
267	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
268	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
269	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
270	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
271	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
272	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
273	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
274	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
275	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
276	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
277	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
278	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
279	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
280	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
281	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
282	C₂²×C₆₇	#585	1	1	yes	characteristic
283	C₄×C₆₇	—	2	8	no	—
284	C₄×C₆₇	—	2	2	no	—
285	C₄×C₆₇	—	2	2	no	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	2	4	no	—
288	C₄×C₆₇	—	2	2	no	—
289	C₄×C₆₇	—	4	4	no	—
290	C₄×C₆₇	—	4	2	no	—
291	C₄×C₆₇	—	4	2	no	—
292	C₄×C₆₇	—	4	2	no	—
293	C₄×C₆₇	—	4	2	no	—
294	C₄×C₆₇	—	4	2	no	—
295	C₄×C₆₇	—	4	1	no	—
296	C₄×C₆₇	—	4	2	no	—
297	C₄×C₆₇	—	4	2	no	—
298	C₄×C₆₇	#554	1	4	yes	—
299	C₄×C₆₇	#557	1	4	yes	—
300	#59666	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
301	C₂×C₆₇	C₂×#554	1	1	yes	characteristic
302	C₂×C₆₇	C₂×#557	1	1	yes	characteristic
303	C₂×C₆₇	#3480	1	1	yes	characteristic
304	C₂×C₆₇	#3502	1	1	yes	characteristic
305	C₂×C₆₇	#3509	1	1	yes	characteristic
306	C₂×C₆₇	#3520	1	1	yes	characteristic
307	C₂×C₆₇	#3547	1	1	yes	characteristic
308	#2802	C₂×C₆₇	1	4	yes	—
309	#2582	C₂×C₆₇	1	2	yes	—
310	#2581	C₂×C₆₇	1	2	yes	—
311	#2571	C₂×C₆₇	1	1	yes	characteristic
312	#2565	C₂×C₆₇	1	2	yes	—
313	#2552	C₂×C₆₇	1	1	yes	characteristic
314	#2550	C₂×C₆₇	1	2	yes	—
315	#2520	C₂×C₆₇	1	4	yes	—
316	#2508	C₂×C₆₇	1	2	yes	—
317	#2507	C₂×C₆₇	1	2	yes	—
318	#2370	C₂×C₆₇	1	2	yes	—
319	#2365	C₂×C₆₇	1	1	yes	characteristic
320	#2365	C₂×C₆₇	1	1	yes	characteristic
321	#2351	C₂×C₆₇	1	2	yes	—
322	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
323	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
324	C₄×#125	C₂×C₆₇	1	1	yes	characteristic
325	C₆₇	#59666	1	1	yes	characteristic, direct product factor, Sylow
326	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
328	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×#125	C₂²×C₆₇	1	2	yes	—
330	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
331	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
332	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
333	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
334	#400	C₂²×C₆₇	1	4	yes	—
335	#400	C₂²×C₆₇	1	4	yes	—
336	#398	C₂²×C₆₇	1	4	yes	—
337	#398	C₂²×C₆₇	1	4	yes	—
338	#398	C₂²×C₆₇	1	4	yes	—
339	#398	C₂²×C₆₇	1	4	yes	—
340	#398	C₂²×C₆₇	1	4	yes	—
341	#395	C₂²×C₆₇	1	4	yes	—
342	#391	C₂²×C₆₇	1	2	yes	—
343	#391	C₂²×C₆₇	1	2	yes	—
344	#391	C₂²×C₆₇	1	2	yes	—
345	#391	C₂²×C₆₇	1	2	yes	—
346	#391	C₂²×C₆₇	1	4	yes	—
347	#389	C₂²×C₆₇	1	1	yes	characteristic
348	#389	C₂²×C₆₇	1	1	yes	characteristic
349	#389	C₂²×C₆₇	1	1	yes	characteristic
350	#389	C₂²×C₆₇	1	2	yes	—
351	#389	C₂²×C₆₇	1	1	yes	characteristic
352	#389	C₂²×C₆₇	1	2	yes	—
353	#389	C₂²×C₆₇	1	2	yes	—
354	#389	C₂²×C₆₇	1	2	yes	—
355	#389	C₂²×C₆₇	1	2	yes	—
356	#389	C₂²×C₆₇	1	2	yes	—
357	#389	C₂²×C₆₇	1	4	yes	—
358	#389	C₂²×C₆₇	1	4	yes	—
359	#389	C₂²×C₆₇	1	4	yes	—
360	#389	C₂²×C₆₇	1	2	yes	—
361	#389	C₂²×C₆₇	1	2	yes	—
362	#389	C₂²×C₆₇	1	2	yes	—
363	#387	C₂²×C₆₇	1	2	yes	—
364	#387	C₂²×C₆₇	1	2	yes	—
365	#387	C₂²×C₆₇	1	4	yes	—
366	#387	C₂²×C₆₇	1	4	yes	—
367	#387	C₂²×C₆₇	1	4	yes	—
368	#387	C₂²×C₆₇	1	4	yes	—
369	#387	C₂²×C₆₇	1	4	yes	—
370	#387	C₂²×C₆₇	1	4	yes	—
371	#384	C₂²×C₆₇	1	1	yes	characteristic
372	#384	C₂²×C₆₇	1	1	yes	characteristic
373	#384	C₂²×C₆₇	1	2	yes	—
374	#382	C₂²×C₆₇	1	1	yes	characteristic
375	#382	C₂²×C₆₇	1	2	yes	—
376	#382	C₂²×C₆₇	1	2	yes	—
377	#382	C₂²×C₆₇	1	4	yes	—
378	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
379	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
380	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
381	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
382	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
383	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
384	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
385	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
386	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
387	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
388	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
389	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
390	C₄³	C₂²×C₆₇	1	1	yes	characteristic
391	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
392	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
393	C₄⋊Q₈	—	2	4	no	—
394	C₄⋊Q₈	Q₈×C₆₇	1	8	yes	complemented
395	#125	—	2	8	no	—
396	#125	—	2	4	no	—
397	C₄×Q₈	—	2	8	no	—
398	C₄×Q₈	—	4	4	no	—
399	C₄×Q₈	—	4	4	no	—
400	C₄×Q₈	—	4	4	no	—
401	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
405	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
415	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
424	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
426	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
427	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
428	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
429	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
430	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
432	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
433	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
434	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
435	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
437	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
439	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
440	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
441	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
442	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
443	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
444	#95	C₂³×C₆₇	1	2	yes	—
445	#95	C₂³×C₆₇	1	2	yes	—
446	#95	C₂³×C₆₇	1	2	yes	—
447	#95	C₂³×C₆₇	1	2	yes	—
448	#95	C₂³×C₆₇	1	2	yes	—
449	#95	C₂³×C₆₇	1	2	yes	—
450	#95	C₂³×C₆₇	1	2	yes	—
451	#95	C₂³×C₆₇	1	4	yes	—
452	#95	C₂³×C₆₇	1	4	yes	—
453	#95	C₂³×C₆₇	1	2	yes	—
454	#95	C₂³×C₆₇	1	2	yes	—
455	#95	C₂³×C₆₇	1	4	yes	—
456	#95	C₂³×C₆₇	1	4	yes	—
457	#95	C₂³×C₆₇	1	2	yes	—
458	#95	C₂³×C₆₇	1	4	yes	—
459	#95	C₂³×C₆₇	1	4	yes	—
460	#95	C₂³×C₆₇	1	4	yes	—
461	#95	C₂³×C₆₇	1	4	yes	—
462	#95	C₂³×C₆₇	1	4	yes	—
463	#95	C₂³×C₆₇	1	4	yes	—
464	C₂×Q₈	—	2	4	no	—
465	C₂×Q₈	—	2	4	no	—
466	C₂×Q₈	—	2	4	no	—
467	C₂×Q₈	—	2	4	no	—
468	C₂×Q₈	—	4	2	no	—
469	C₂×Q₈	C₂×Q₈×C₆₇	1	8	yes	—
470	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
471	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
472	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
474	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
475	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
476	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
478	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
479	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
482	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
487	C₄⋊C₄	—	2	8	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	8	no	—
490	C₄⋊C₄	—	2	8	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	4	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	4	2	no	—
504	C₄⋊C₄	—	4	2	no	—
505	C₄⋊C₄	—	4	4	no	—
506	C₄⋊C₄	—	4	4	no	—
507	C₄⋊C₄	—	4	2	no	—
508	C₄⋊C₄	—	4	2	no	—
509	C₄⋊C₄	—	4	2	no	—
510	C₄⋊C₄	—	4	2	no	—
511	C₄⋊C₄	—	4	4	no	—
512	C₄⋊C₄	—	4	4	no	—
513	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
514	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
515	C₄⋊C₄	#41×C₆₇	1	4	yes	—
516	C₄²	—	2	8	no	—
517	C₄²	—	2	4	no	—
518	C₄²	—	2	2	no	—
519	C₄²	—	2	4	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	2	4	no	—
522	C₄²	—	2	2	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	2	4	no	—
525	C₄²	—	4	4	no	—
526	C₄²	—	4	4	no	—
527	C₄²	—	4	1	no	—
528	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
529	C₄²	#41×C₆₇	1	4	yes	—
530	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
531	Q₈	—	2	16	no	—
532	Q₈	—	4	8	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
573	C₂×C₄	C₂²×Q₈×C₆₇	1	8	yes	—
574	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
575	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
576	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
577	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
578	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
579	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
580	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
581	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
582	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
583	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
584	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
585	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
586	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
587	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
588	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
589	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
590	C₂²	#585×C₆₇	1	1	yes	characteristic
591	C₄	—	2	8	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	2	4	no	—
596	C₄	—	2	2	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	2	no	—
600	C₄	—	4	2	no	—
601	C₄	—	4	4	no	—
602	C₄	—	4	2	no	—
603	C₄	—	4	2	no	—
604	C₄	—	4	2	no	—
605	C₄	—	4	2	no	—
606	C₄	#554×C₆₇	1	4	yes	—
607	C₄	#557×C₆₇	1	4	yes	—
608	C₂	C₂×#554×C₆₇	1	1	yes	characteristic
609	C₂	C₂×#557×C₆₇	1	1	yes	characteristic
610	C₂	C₆₇×#3480	1	1	yes	characteristic
611	C₂	C₆₇×#3502	1	1	yes	characteristic
612	C₂	C₆₇×#3509	1	1	yes	characteristic
613	C₂	C₆₇×#3520	1	1	yes	characteristic
614	C₂	C₆₇×#3547	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
86. C₄⋊Q₈×C₆₇	Q₈	semidirect
300. #59666	C₆₇	direct (N × H)
325. C₆₇	#59666	direct (N × H)
394. C₄⋊Q₈	Q₈×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#2802	2
C₆₇×#2582	2
C₆₇×#2581	2
C₆₇×#2571	2
C₆₇×#2565	2
C₆₇×#2552	2
C₆₇×#2550	2
C₆₇×#2520	2
C₆₇×#2508	2
C₆₇×#2507	2
C₆₇×#2370	2
C₆₇×#2365	2
C₆₇×#2351	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#125×C₆₇	2
#59666	67

Subgroups — C67×#59666 #30738917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59666 #30738917